Encoding method and encoder for (n,n(n-1),n-1) permutation group code in communication modulation system

ABSTRACT

The present disclosure provides an encoding method and an encoder for a (n, n(n−1), n−1) permutation group code in a communication modulation system, in which 2k k-length binary information sequences are mapped to 2k n-length permutation codeword signal points in a n-dimensional modulation constellation Γn. The constellation Γn with the coset characteristics is formed by selecting 2k n-length permutation codewords from n(n−1) permutation codewords of a code set Pn,xi of the (n, n(n−1), n−1) permutation group code based on coset partition. The constellation Γn is a coset code in which 2k1 cosets are included and each coset includes 2k2 permutation codewords, where k=k1+k2, and 2k≤n(n−1). The present disclosure utilizes the coset characteristics to realize one-to-one correspondence mapping of the binary information sequence set to the permutation code constellation, so that the time complexity of executing the encoder is at most the linear complexity of the code length n.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the priority benefit of China application serialno. 201910169263.8, filed on Mar. 6, 2019. The entirety of theabove-mentioned patent application is hereby incorporated by referenceherein and made a part of this specification.

TECHNICAL FIELD

The present disclosure relates to a technical field of channel codingmodulation in communication transmission, and more particularly to anencoding method and an encoder for a (n, n(n−1), n−1) permutation groupcode (PGC) in a communication modulation system.

DESCRIPTION OF THE RELATED ART

In the wireless signal transmission in multi-user communication, thereare not only multipath fading, but also multi-user interference. TheCDMA-based multiple access scheme in the 3G mobile communication and theOFDM-based multiple access scheme in the 4G mobile communication bothhas the strong capability for resisting multipath fading and multi-userinterference, but feature a large system delay, which is difficult tomeet the needs of the 5G in specific application areas. To this end, amulti-access coding modulation scheme with ultra-low delay andultra-reliable reliability is proposed. The basic principle of thescheme is that a (n, n(n−1), n−1) permutation group code is utilized tocontrol the carrying signal, and in this process, time diversity andfrequency diversity are introduced at the same time, so that the systemstill has the strong capability for resisting multipath fading andmulti-user interference in a case of performing a reduced-complexityoperation.

A patent application entitled “CONSTRUCTION METHOD FOR (n, n(n−1), n−1)PERMUTATION GROUP CODE BASED ON COSET PARTITION AND CODEBOOK GENERATORTHEREOF” was filed in the China National Intellectual PropertyAdministration (CNIPA) on Jan. 27, 2016 with the application No.:201610051144.9. In addition, a patent application entitled “CONSTRUCTIONMETHOD FOR (n, n(n−1), n−1) PERMUTATION GROUP CODE BASED ON COSETPARTITION AND CODEBOOK GENERATOR THEREOF” was filed in the United StatesPatent and Trademark Office (USTPO) with the application Ser. No.:15/060,111, and has been granted patent right.

As far as the current research status is concerned, there is noeffective mapping encoding algorithm for the permutation group code andspecific executable scheme for the corresponding encoder in the PGC-MFSKcoded modulation transceiver system. In particular, due to the lack ofalgebraic encoding and decoding schemes for the permutation code, arandom permutation code is used in most of the research results onpermutation code applications.

SUMMARY

In view of the algebraic structure of the (n, n(n−1), n−1) permutationgroup code based on coset partition, the present disclosure discloses anencoding method and an encoder for a (n, n(n−1), n−1) permutation groupcode based on coset partition in a communication modulation system, inwhich a k-length binary information sequence is mapped to a codeword inthe (n, n(n−1), n−1) permutation group code, and a set of 2^(k) k-lengthbinary information sequences is mapped to a code set of a permutationgroup code with a code length of n, a minimum distance of n−1, acardinality of n(n−1) and an error-correcting capability of d−1=n−2,where 2^(k)≤n(n−1). This requires that 2^(k) codewords are selected fromthe n(n−1) codewords of the code set of the permutation group code tomatch the 2^(k) k-length binary information sequences one by one. Sincethe (n, n(n−1), n−1) permutation group code has the coset partitionstructure characteristics, an encoding method and an encoder for thecoset structure can be designed to form a time-diversity andfrequency-diversity channel access technology with ultra-low delay.

In order to achieve the above objective, according to an aspect of thepresent disclosure, there is provided an encoding method for a (n,n(n−1), n−1) permutation group code in a communication modulationsystem, wherein the encoding method maps a k-length binary informationsequence to a n-length permutation codeword in a signal constellationformed by a (n, n(n−1), n−1) permutation group code based on cosetpartition, and comprises the following steps.

Step 1: constructing a (n, n(n−1), n−1) permutation group code, whereinwhen n is a prime number, the (n, n(n−1), n−1) permutation group codecontains n(n−1) permutation codewords, each permutation codewordcontains n code elements, a minimum Hamming distance between any twopermutation codewords is n−1, and a code set P_(n,x) _(i) of the (n,n(n−1), n−1) permutation group code is obtained by followingexpressions:

$\begin{matrix}{P_{n,x_{i}} = {{C_{n}L_{n,x_{i}}} = \left\{ {\left. {c_{i} \circ l_{j}} \middle| {c_{i} \in C_{n}} \right.,{l_{j} \in L_{n.x_{i}}},{i \in Z_{n}},{j \in Z_{n - 1}}} \right\}}} & (1) \\{= {\left\{ {\left( t_{rn} \right)^{n - 1}L_{n,x_{i}}} \right\} = \left\{ {\left( t_{l1} \right)^{n - 1}L_{n,x_{i}}} \right\}}} & (2)\end{matrix}$where P_(n,x) _(i) is obtained by an operator “∘” composition operationof a special cyclic subgroup C_(n) with a cardinality of |C_(n)|=n and alargest single fixed point subgroup L_(n,x) _(i) , c_(i) represents anelement in C_(n), l_(j) represents an element in L_(n,x) _(i) , Z_(n)represents a positive integer finite domain expressed by Z_(n)={1,2, . .. , n}, Z_(n-1) represents a positive integer finite domain expressed byZ_(n-1)={1,2, . . . , n−1}, and the code set P_(n,x) _(i) has acardinality of |P_(n,x) _(i) |=n(n−1).

The largest single fixed point subgroup L_(n,x) _(i) is obtained by anexpression (3):L _(n,x) _(i) ={a(l _(1,x) _(i) −x _(i))+x _(i) |a∈Z _(n-1) ,x _(i) ∈Z_(n) ,l _(1,x) ₁ =[1 . . . n]}  (3)where when n is a prime number, the largest single fixed point subgrouphas a cardinality of |L_(n,x) _(i) |=n−1; x_(i)∈Z_(n) indicates that thei-th code element of each of all n−1 permutation codewords in L_(n,x)_(i) is a fixed point, and the other code elements are non-fixed points;and for x_(i)=i and x_(i), i∈Z_(n), there are n fixed points in all,respectively corresponding to n largest single fixed point subgroupsL_(n,1), L_(n,2), . . . , L_(n,n).

P_(n,x) _(i) =C_(n)L_(n,x) _(i) indicates that the code set P_(n,x) _(i)is composed of n−1 cosets C_(n)l_(1,x) _(i) , C_(n)l_(2,x) _(i) , . . ., C_(n)l_(n-1,x) _(i) of C_(n); in the expression (2), C_(n) is replacedby a cyclic-right-shift operator function (t_(rn))^(n-1) or acyclic-left-shift operator function (t_(l1))^(n-1) to act on L_(n,x)_(i) , so that the n−1 cosets of C_(n) are equivalently expressed as n−1cyclic-left-shift orbits {(t_(l1))^(n-1)l_(1,x) _(i) },{(t_(l1))^(n-1)l_(2,x) _(i) }, . . . , {(t_(l1))^(n-1)l_(n-1,x) _(i) }or cyclic-right-shift orbits {(t_(rn))^(n-1)l_(1,x) _(i) },{(t_(rn))^(n-1)l_(2,x) _(i) }, . . . , {(t_(rn))^(n-1)l_(n-1,x) _(i) }.

For an equivalent operation of {(t_(rn))^(n-1)L_(n,x) _(i) } and{(t_(l1))^(n-1)L_(n,x) _(i) }, a coset leader or orbit leader arraycomposed of n−1 permutation codewords is first calculated by theexpression (3) for calculating L_(n,x) _(i) , and then all n(n−1)codewords of the code set P_(n,x) _(i) are calculated by the expression(2); for each orbit, a permutation codeword l_(j,x) _(i) of L_(n,x) _(i)is input to a cyclic shift register to perform n−1 cyclic-left-shiftoperations, equivalent to performing {(t_(l1))^(n-1)l_(j,x) _(i) }, orto perform n−1 cyclic-right-shift operations, equivalent to performing{(t_(rn))^(n-1)l_(j,x) _(i) }, thereby generating n permutationcodewords; and n(n−1) permutation codewords are generated by n−1 orbits.

Step 2: selecting 2^(k) permutation codewords from the n(n−1)permutation codewords of the code set P_(n,x) _(i) to form a signalconstellation Γ_(n) such that Γ_(n) still has coset characteristics,that is, Γ_(n) contains the 2^(k) permutation codewords, and ispartitioned into 2^(k) ¹ cosets, and each coset contains 2^(k) ²permutation codewords, where k=k₁+k₂, 2^(k)≤n(n−1), 2^(k) ¹ ≤n−1, 2^(k)² ≤n, and an exact value of k is k=└log₂ n(n−1)┘.

Step 3: defining a mapping function φ: H_(k)→Γ_(n) by a function π=φ(h),and mapping, by the mapping function φ, a k-length binary informationsequence h=[h₁h₂ . . . h_(k)] in a set H_(k) of 2^(k) k-length binaryinformation sequences to a signal point π=[a₁a₂ . . . a_(n)] in thesignal constellation Γ_(n) composed of the 2^(k) n-length permutationcodewords, where π∈Γ_(n), h∈H_(k), h₁, h₂, . . . , h_(k)∈Z₂={0,1}, anda₁, a₂, . . . , a_(n)∈Z_(n).

In an alternative embodiment, the signal constellation Γ_(n) with thecoset characteristics constitutes a coset code, and the step 2specifically includes the following.

For any prime number n>1, the code set P_(n,x) _(i) is a subgroup of asymmetric group S_(n), and all permutation codewords of P_(n,x) _(i) arepositive integer vectors, which are regarded as discrete lattice pointsin an n-dimensional real Euclidean space

^(n); all valid signal points of the signal constellation Γ_(n) areselected from the subgroup P_(n,x) _(i) or finite lattice P_(n,x) _(i)of the symmetric group S_(n); Γ_(n) has a cardinality of|Γ_(n)|=|H_(k)|=2^(k), and has the same coset structure as P_(n,x) _(i)and cosets of Γ_(n) have the same size, which is smaller than that ofP_(n,x) _(i) .

For a sub-lattice or sub-group C_(n) of the code set P_(n,x) _(i) ,i.e., a subset of n(n−1) permutation codewords of P_(n,x) _(i) , thesub-lattice C_(n) itself is an n-dimensional lattice or a set ofn-dimensional permutation vectors each containing n permutationcodewords, and induces a partition P_(n,x) _(i) /C_(n) of P_(n,x) _(i) ,which partitions P_(n,x) _(i) into |P_(n,x) _(i) /C_(n)| cosets ofC_(n), where |P_(n,x) _(i) /C_(n)|=|L_(n,x) _(i) |=n−1; in the signalconstellation Γ_(n)∈P_(n,x) _(i) , when a coset of C_(n) and a latticepoint of the coset are respectively indexed by two binary informationsequences, the order of this partition is expressed as a power of 2, thenumber of cosets of this division is 2^(k) ¹ , and each coset is indexedby a k₁-length information sequence; in each coset, the number of validlattice points is also a power of 2, then the number of signal pointscontained in each coset is 2^(k) ² , and each signal point in each cosetis indexed by a k₂-length information sequence, where k=k₁+k₂.

The step 3 specifically includes the following.

The k-length binary information sequence is separated into twoindependent binary information sequences: a k₁-length informationsequence corresponding to high-level k₁-bit of the k-length binaryinformation sequence, and a k₂-length information sequence correspondingto low-level k₂-bit of the k-length binary information sequence; thek₁-length information sequence is used for indexing a coset in |P_(n,x)_(i) /C_(n)|=|L_(n,x) _(i) |=n−1 cosets, that is, for selecting a cosetin the 2^(k) ¹ =n−1 cosets in Γ_(n); the k₂-length information sequenceis used for indexing a codeword in the selected coset, that is, forselecting a codeword in the 2^(k) ² <n codewords of the selected cosetto be transmitted to a channel, thereby obtaining one-to-one mappingnumberings between the 2^(k) k-length binary information sequences andthe 2^(k) n-length permutation codewords of the signal constellationΓ_(n).

In accordance with another aspect of the present disclosure, there isfurther provided an encoder for a (n, n(n−1), n−1) permutation groupcode in a communication modulation system, wherein the encoder maps ak-length binary information sequence to a n-length permutation codewordin a signal constellation formed by a (n, n(n−1), n−1) permutation groupcode based on coset partition, the encoder comprises a k-lengthinformation sequence splitter (also called a bit splitter) D, a cosetselector and an intra-coset permutation codeword selector.

The k-length information sequence splitter D is configured to inputs ak-length binary information sequence and output two informationsequences: a k₁-length information sequence corresponding to high-levelk₁-bit of the input k-length binary information sequence, and ak₂-length information sequence corresponding to low-level k₂-bit of theinput k-length binary information sequence, where k=k₁+k₂.

The coset selector is configured to select a coset by taking k₁-lengthinformation sequences as indices of n−1 cosets, in which 2^(k) ¹k₁-length information sequences respectively correspond to 2^(k) ¹binary index labels for the cosets, 2^(k) ¹ ≤n−1, and each binary indexlabel is used for selecting a coset from 2^(k) ¹ cosets; when n is aprime number, 2^(k) ¹ =n−1; the cosets are as follows: 2^(k) permutationcodewords are selected from n(n−1) permutation codewords of a code setP_(n,x) _(i) to form a signal constellation Γ_(n) such that Γ_(n) stillhas coset characteristics, that is, Γ_(n) contains 2^(k) permutationcodewords, and is partitioned into 2^(k) ¹ cosets, and each cosetcontains 2^(k) ² permutation codewords, where k=k₁+k₂, 2^(k)≤n(n−1),2^(k) ¹ ≤n−1, 2^(k) ² ≤n, and an exact value of k is k=└log₂ n(n−1)┘.

The intra-coset permutation codeword selector is configured to select apermutation codeword by taking k₂-length information sequences asindices of n permutation codewords in the selected coset, in which the2^(k) ² k₂-length information sequences respectively correspond to 2^(k)² binary index labels for the n permutation codewords in the selectedcoset, 2^(k) ² ≤n and each binary index label is used for selecting apermutation codeword from 2^(k) ² permutation codewords in the selectedcoset; when n is a prime number, 2^(k) ² permutation codewords areselected from n permutation codewords in a coset of C_(n) as needed toform a coset of the constellation Γ_(n), that is, any n−2^(k) ²permutation codewords are required to be discarded in each coset ofC_(n).

In an alternative embodiment, the encoder is one of the following threeencoders: a U₁-V₁ type encoder with all permutation codewords of theconstellation Γ_(n) stored in an n-dimensional read only memory (ROM), aU₁-V₂ type encoder with a part of permutation codewords of theconstellation Γ_(n) stored in an n-dimensional ROM, and a U₂-V₂ typeencoder with the constellation Γ_(n) independent of an n-dimensionalROM, wherein U₁ and U₂ represent two different types of coset selectors,and V₁ and V₂ represent two different types of intra-coset permutationcodeword selectors.

In a case where the coset selector is a U₁ type coset selector, thecoset selector includes two parts: an address generator of mapping ak₁-length information sequence to a coset leader permutation codeword,in which when k₁ is input, an address in a n-dimensional ROM is output;and a storage structure of 2^(k) ¹ coset leader permutation codewords inthe n-dimensional ROM.

In a case where the coset selector is a U₂ type coset selector, thecoset selector includes two parts: a mapper of mapping a k₁-lengthinformation sequence to a parameter a; and a coset leader permutationcodeword generator.

In a case where the intra-coset permutation codeword selector is a V₁type intra-coset permutation codeword selector, the intra-cosetpermutation codeword selector includes two parts: an address generatorof mapping a k₂-length information sequence to an intra-cosetpermutation codeword, in which when k₂ is input, an address in ann-dimensional ROM is output; and a storage structure of 2^(k) ²intra-coset permutation codewords in the n-dimensional ROM.

In a case where the intra-coset permutation codeword selector is a V₂type intra-coset permutation codeword selector, the intra-cosetpermutation codeword selector includes two parts: a decrement counterfor k₂-length information sequence; and a cyclic shift register with twoswitches configured to perform a cyclic-left-shift or cyclic-right-shiftoperation.

In an alternative embodiment, the U₁-V₁ type encoder includes thek-length information sequence splitter D, the address generator ofmapping the k₁-length information sequence to the coset leaderpermutation codeword, the address generator of mapping the k₂-lengthinformation sequence to the intra-coset permutation codeword, and astorage structure of all 2^(k) permutation codewords of theconstellation Γ_(n) in the n-dimensional ROM.

For a structure of the address generator of mapping the k₁-lengthinformation sequence to the coset leader permutation codeword, there isa one-to-one correspondence between 2^(k) ¹ k₁-length binary informationsequences and coset leaders of respective cosets; each coset leader isdetermined by a largest single fixed point subgroup L_(n,x) _(i) , and|L_(n,x) _(i) | coset leader permutation codewords are calculated byL_(n,x) _(i) ={a(l_(1,x) _(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n),l_(1,x) _(i) =[1 . . . n]}; 2^(k) ¹ coset leader permutation codewordsare selected from the |L_(n,x) _(i) | coset leader permutation codewordsto be stored in the n-dimensional ROM, and a storage address of each ofthe 2^(k) ¹ coset leader permutation codewords in the n-dimensional ROMis recorded, thereby forming the address generator; when n is a primenumber, |L_(n,x) _(i) |=n−1=2^(k) ¹ ; and the address generator ofmapping the k₁-length information sequence to the coset leaderpermutation codeword inputs a k₁-length information sequence and outputsan address of a selected coset leader permutation codeword.

For a structure of the address generator of mapping the k₂-lengthinformation sequence to the intra-coset permutation codeword, aone-to-one correspondence between 2^(k) ² k₂-length information sequenceand permutation codewords in each coset is established; n−2^(k) ²permutation codewords are discarded in each coset of the code setP_(n,x) _(i) to form the constellation Γ_(n); 2^(k) ² permutationcodewords of each coset of the constellation Γ_(n) are sequentiallystored in the n-dimensional ROM, with a storage address of therespective coset leader permutation codeword followed by storageaddresses of the respective permutation codewords, and a mappingfunction between the 2^(k) ² k₂-length information sequence and storageaddresses of the 2^(k) ² permutation codewords of each coset of theconstellation Γ_(n) in the n-dimensional ROM is established; and theaddress generator of mapping the k₂-length information sequence to theintra-coset permutation codeword inputs a k₂-length information sequenceand outputs an address of a selected intra-coset permutation codeword.

For a storage structure of all 2^(k) permutation codewords of theconstellation Γ_(n) in the n-dimensional ROM, the constellation Γ_(n) ispartitioned into 2^(k) ¹ cosets, and each coset contains 2^(k) ²permutation codeword; each permutation codeword of the largest singlefixed point subgroup is calculated, and a storage address of each cosetleader permutation codeword in the n-dimensional ROM is determined;2^(k) ¹ storage addresses are stored in a register of the addressgenerator of mapping the k₁-length information sequence to the cosetleader permutation codeword to select respective coset leaderpermutation codewords; when n is a prime number, 2^(k) ¹ =n−1; 2^(k) ²permutation codewords in each coset are obtained by a cyclic-left-shiftor cyclic-right-shift composition operator acting on each coset leaderpermutation codeword, that is, calculating a set (t_(l1))^(n-1)L_(n,x)_(i) } or {(t_(rn))^(n-1)L_(n,x) _(i) }, and stored in the n-dimensionalROM, with a storage address of the respective coset leader permutationcodeword followed by storage addresses of the respective permutationcodewords; and an address corresponding to the k₂-length informationsequence is determined by the k₂-length information sequence itself andthe address of the coset leader permutation codeword.

In an alternative embodiment, the U₁-V₂ type encoder includes thek-length information sequence splitter D, the address generator ofmapping the k₁-length information sequence to the coset leaderpermutation codeword, the storage structure of the 2^(k) ¹ coset leaderpermutation codewords in the n-dimensional ROM, the decrement counterfor the k₂-length information sequence, and the cyclic shift registerwith two switches.

For the storage structure of the 2^(k) ¹ coset leader permutationcodewords in the n-dimensional ROM, when 2^(k) ¹ permutation codewordsof L_(n,x) _(i) are stored in the n-dimensional ROM, each linerepresents a storage word in the n-dimensional ROM, and each permutationcodeword occupies a storage word; all |L_(n,x) _(i) | coset leaderpermutation codewords are generated by L_(n,x) _(i) ={a(l_(1,x) _(i)−x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1 . . . n]}, andthe 2^(k) ¹ permutation codewords are arbitrarily selected from the|L_(n,x) _(i) | coset leader permutation codewords, and sequentiallystored in the n-dimensional ROM in an order of a=1,2, . . . , |L_(n,x)_(i) |; and in a case of a read control signal Rd=1, upon the arrival ofa cp clock pulse, the n-dimensional ROM outputs a permutation codeworddesignated by an input address in parallel or serial.

For a structure of the decrement counter for the k₂-length informationsequence, the decrement counter for the k₂-length information sequenceinputs a k₂-length information sequence corresponding to low-levelk₂-bit of the k-length information sequence, and the k₂-lengthinformation sequence is stored to a u cyclic shift register in thedecrement counter to perform a cycle-minus-one operation; when u≠0, thedecrement counter outputs a high-level signal to control a switch 1 tobe closed and a switch 2 to be opened; and when u=0, the decrementcounter outputs a low-level signal to control the switch 1 to be openedand the switch 2 to be closed; and the switch 1 is controlled to performa cyclic shift operation of the cyclic shift register, and the switch 2is controlled to perform a serial output operation of the cyclic shiftregister.

For the cyclic shift register, when the switch 1 is controlled to beclosed, the cyclic shift register performs a cyclic-left-shift orcyclic-right-shift operation on a permutation codeword stored therein toobtain a new permutation codeword, and such cyclic shift operation isperformed k₂ times, until u is decremented from u≠0 to u=0 through thecycle-minus-one operation, thereby forming a decoded codeword in thecyclic shift register; then the switch 1 is controlled to be opened, andthe switch 2 is controlled to be closed, so as to serially output adecoded codeword.

For a working process of the U₁-V₂ type encoder with the part ofpermutation codewords of the constellation Γ_(n) stored in then-dimensional ROM, the information sequence splitter D inputs a k-lengthinformation sequence and partitions it into a k₁-length informationsequence corresponding to high-level k₁-bit and a k₂-length informationsequence corresponding to low-level k₂-bit; the k₁-information sequenceis mapped to an address of a coset leader permutation codeword in then-dimensional ROM, and the address generator outputs the address toselect the coset leader permutation codeword; the selected coset leaderpermutation codeword is input in parallel from the n-dimensional ROM tothe cyclic shift register through a system bus, and under the control ofthe decrement counter for the k₂-length information sequence, ahigh-level signal is output in a case of u≠0, so that the switch 1 isclosed and the cyclic shift register performs a cyclic-left-shift orcyclic-right-shift operation; the cyclic-left-shift orcyclic-right-shift operation is performed once for each cycle-minus-oneoperation of the decrement counter, until u is decremented 0, and then alow-level signal is output, so that the switch 1 is opened, the switch 2is closed, and the cyclic shift register stops the cyclic-left-shift orcyclic-right-shift operation, but performs a left-shift-output operationto serially output a decoded codeword.

In an alternative embodiment, the U₂-V₂ type encoder includes: thek-length information sequence splitter D, the mapper of mapping thek₁-length information sequence to the parameter a, the structure of thecoset leader permutation codeword generator, the decrement counter fork₂-length information sequence, and the cyclic shift register with twoswitches.

For the mapper of mapping the k₁-length information sequence to theparameter a, there is a one-to-one correspondence between 2^(k) ¹k₁-length information sequences and values of the parameter a in thecalculation expression L_(n,x) _(i) ={a(l_(1,x) _(i)−x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1 . . . n]} of thelargest single fixed point subgroup; all |L_(n,x) _(i) | coset leaderpermutation codewords are generated by the above calculation expression,and 2^(k) ¹ permutation codewords are selected from the |L_(n,x) _(i) |coset leader permutation codewords to form 2^(k) ¹ coset leaderpermutation codewords of the constellation Γ_(n), so that a k₁-lengthinformation sequence determines a coset leader permutation codeword ofthe constellation Γ_(n).

In general, by comparing the above technical solution of the presentinventive concept with the prior art, the present disclosure has thefollowing beneficial effects.

In the encoding method for a (n, n(n−1), n−1) permutation group codebased on coset partition proposed by the present disclosure, aone-to-one correspondence mapping between a binary sequence and acodeword is achieved by utilizing the coset characteristics, and a firstpermutation codeword of each code set can be obtained by a simple modulon operation instead of a complex composition operation. After firstpermutation codewords of all code sets are determined, other permutationcodewords in the code set can be obtained by a cyclic shift register. Asa multi-ary error-correcting code class, the permutation group code hasan error-correcting capability of d−1, which is twice theerror-correcting capability of └(d−1)/2┘ of the conventional multi-aryerror-correcting code class. When combined with the MFSK modulationtechnique, the receiver can perform demodulation by a simplenon-coherent constant envelope demodulation technique. The reliabilityof signal transmission can be guaranteed in the interference channelwhere both multi-frequency noise and deep fading exist.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a corresponding relation diagram of a binary informationsequence and a permutation codeword in an encoding method according tothe present disclosure.

FIG. 2 is a schematic diagram showing an encoding process of theencoding method according to the present disclosure.

FIG. 3 is a block diagram showing the basic principle of a mappingencoder for a constellation Γ_(n) according to the present disclosure.

FIG. 4 shows a general architecture of the mapping encoder for theconstellation Γ_(n) according to the present disclosure.

FIG. 5 shows architecture of a mapping encoder with all permutationcodewords of the constellation Γ_(n) stored in an n-dimensional ROM (aU₁-V₁ type encoder) according to the present disclosure.

FIG. 6 shows architecture of a mapping encoder with a part ofpermutation codewords of the constellation Γ_(n) stored in ann-dimensional ROM (a U₁-V₂ type encoder) according to the presentdisclosure.

FIG. 7 shows a storage structure of 2^(k) ¹ coset leader permutationcodewords in the n-dimensional ROM according to the present disclosure.

FIG. 8 shows architecture of a mapping encoder with the constellationΓ_(n) independent of an n-dimensional ROM (a U₂-V₂ type encoder)according to the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

For clear understanding of the objectives, features and advantages ofthe present disclosure, detailed description of the present disclosurewill be given below in conjunction with accompanying drawings andspecific embodiments. It should be noted that the embodiments describedherein are only meant to explain the present disclosure, and not tolimit the scope of the present disclosure. Furthermore, the technicalfeatures related to the embodiments of the present disclosure describedbelow can be mutually combined if they are not found to be mutuallyexclusive.

Basic Principles

Basic principles of a mapping encoding method of a (n, n(n−1), n−1)permutation group code based on coset partition according to the presentdisclosure are described below.

Assuming that code symbols can take values in a positive integer finitedomain Z_(n)={1,2, . . . , n} or an integer finite domain Z_(n)⁰={0,1,2, . . . , n−1}, but a description with values mainly taken inZ_(n)={1,2, . . . , n} is given below, and the result also applies tothe case where values are taken in Z_(n) ⁰={0,1,2, . . . , n−1}.

Calling a set formed by all n! permutations of n elements in Z_(n) asymmetric group S_(n)={π₁, . . . , π_(k), . . . , π_(n!)}, an element inS_(n) may be represented by a permutation vector π_(k)=[a₁ . . . a_(i) .. . a_(n)]. All elements of a permutation are different and representedby a₁, . . . , a_(i), . . . , a_(n)∈Z_(n). Degree (dimension, size) of apermutation is |π_(k)|=n, and cardinality (order) of the symmetric groupis |S_(n)|=n!. Let π₀=e=[a₁a₂ . . . a_(n)]=[12 . . . n] represent anidentity element of the symmetric group S_(n). A general permutationgroup code is defined as a subgroup of the symmetric group S_(n), andexpressed as (n, μ, d)-PGC, where n represents a code length, μrepresents the maximum cardinality (maximum size) of the code set, and drepresents a minimum Hamming distance between any two permutationcodewords in the code set. For example, a (n, n(n−1), n−1) permutationgroup code is a group code with a code length of n, a cardinality ofn(n−1) and a minimum Hamming distance of n−1.

Coset Partition Structure of a (n, n(n−1), n−1) Permutation Group Code

The existing published research results show that a code set P_(n,x)_(i) of a (n, μ, d) permutation group code (n>1) can be equivalentlyobtained by calculating each codeword by the following three methods.

$\begin{matrix}\begin{matrix}{P_{n,x_{i}} = {\left\{ {p_{1},p_{2},\ldots\;,p_{n{({n - 1})}}} \right\} x_{i}}} \\{= {{S_{n}\bigcap\left\{ {C_{n} \circ L_{n,x_{i}}} \right\}} = {S_{n}\bigcap\left\{ \left\{ {c_{i} \circ l_{j,x_{i}}} \right\}_{i = 1}^{n} \right\}_{j = 1}^{n - 1}}}}\end{matrix} & (1) \\{{= {S_{n}\bigcap\left\{ {\left. {{a\left( {p_{1,x_{i}} - x_{i}} \right)} + x_{i} + b} \middle| {a \in Z_{n - 1}} \right.,x_{i},{b \in Z_{n}},{p_{1,x_{i}} = \left\lbrack {1\mspace{11mu}\ldots\mspace{11mu} n} \right\rbrack}} \right\}}}\;} & \left( {2\text{-}1} \right) \\{\mspace{56mu}{= {S_{n}\bigcap\left\{ {L_{n,x_{i}} + b} \middle| {b \in Z_{n}} \right\}}}} & \left( {2\text{-}2} \right) \\{\mspace{56mu}{= {{S_{n}\bigcap\left\{ {\left( t_{rn} \right)^{n - 1}L_{n,x_{i}}} \right\}} = {S_{n}\bigcap\left\{ {\left( t_{l\; 1} \right)^{n - 1}L_{n,x_{i}}} \right\}}}}} & (3)\end{matrix}$where the expression (1) represents a first method for generating thecode set P_(n,x) _(i) , indicating that the code set P_(n,x) _(i)obtained by an operator “∘” composition operation of two smallersubgroups (i.e., a special cyclic subgroup C_(n) with a cardinality of|C_(n)|=n, and a largest single fixed point subgroup L_(n,x) _(i) ); theexpressions (2-1) and (2-2) represent a second method for generating thecode set P_(n,x) _(i) , indicating that each permutation codeword in thecode set P_(n,x) _(i) can be calculated by affine transformationf_(a,b)(p_(1,x) _(i) )=a(p_(1,x) _(i) −x_(i))+x_(i)+b; and theexpression (3) represents a third method for generating the code setP_(n,x) _(i) , indicating that P_(n,x) _(i) can be obtained by using acyclic-right-shift operator (t_(rn))^(n-1) or an equivalentcyclic-left-shift operator (t_(l1))^(n-1) to act on the largest singlefixed point subgroup L_(n,x) _(i) (that is, the cyclic shift operationis performed for n−1 times). The fixed point x_(i)∈Z_(n) indicates thatall permutation vectors in the largest single fixed point subgroupL_(n,x) _(i) contain a fixed point x_(i) and the other code elements arenon-fixed points.

When n is a non-prime number, all the above-mentioned sets formed by thecurly braces {⋅} require an intersection operation with the symmetricgroup S_(n) to guarantee that each element in P_(n,x) _(i) is apermutation vector. This is because for any non-prime number n and alla∈Z_(n-1), when a does not satisfy GCD(a, n)=1, the scalingtransformation f_(a,x) _(i) (l_(1,x) _(i) )=a(l_(1,x) _(i) −x_(i))+x_(i)cannot guarantee that all vectors in the set L_(n,x) _(i) ={a(l_(1,x)_(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i), i∈Z_(n), l_(1,x) _(i) =[1 . . . n]}are permutation vectors, that is, |L_(n,x) _(i) |<n−1. If and only if nis a prime number, all the sets formed by the curly braces {⋅} canguarantee that P_(n,x) _(i) generated by three methods all containn(n−1) permutation codes without the need of an intersection operationwith the symmetric group S_(n), that is, L_(n,x) _(i) is a (n, n−1, n−1)permutation group or |L_(n,x) _(i) |=n−1. GCD(a, n) represents agreatest common divisor between a∈Z_(n-1) and n, and GCD(a, n)=1indicates that a∈Z_(n-1) and n are mutually exclusive.

For any n>1, in the above three methods for generating the code setP_(n,x) _(i) of the (n, μ, d) permutation group code, the expression (3)P_(n,x) _(i) =S_(n)∩{(t_(rn))^(n-1)L_(n,x) _(i)}=S_(n)∩{(t_(l1))^(n-1)L_(n,x) _(i) } has the lowest computationalcomplexity. For the convenience of description, the code set ishereinafter calculated by P_(n,x) _(i) =S_(n)∩{(t_(l1))^(n-1)L_(n,x)_(i) }. If and only if n is a prime number, the code set P_(n,x) _(i) isa (n, n(n−1), n−1) permutation group code, and can be simplisticallycalculated by P_(n,x) _(i) ={(t_(rn))^(n-1)L_(n,x) _(i)}={(t_(l1))^(n-1)L_(n,x) _(i) }.

The coset characteristics of the code set P_(n,x) _(i) can be summarizedas follows:

1) For any fixed point x_(i)∈Z_(n), the code set P_(n,x) _(i) iscomposed of n−1 C_(n) cosets C_(n)l_(1,x) _(i) , C_(n)l_(2,x) _(i) , . .. , C_(n)l_(n-1,x) _(i) , each coset including n codewords.

2) For any fixed point x_(i)∈Z_(n), the code set P_(n,x) _(i) can alsobe regarded as being composed of n−1 orbits {(t_(l1))^(n-1)l_(1,x) _(i)}, {(t_(l1))^(n-1)l_(2,x) _(i) }, . . . , {(t_(l1))^(n-1)l_(n-1,x) _(i)} or {(t_(rn))^(n-1)l_(1,x) _(i) }, {(t_(rn))^(n-1)l_(2,x) _(i) }, . . ., {(t_(rn))^(n-1)l_(n-1,x) _(i) }, each orbit including n codewords.

Example 1: let n=7, which is a prime number, and let a fixed pointx_(i)=x₇=7. The computation expression of L_(7,7) is L_(7,7)={al_(1,x)_(i) |a∈Z₆, l_(1,x) _(i) =[1234567]}, and thus a largest single fixedpoint subgroup in which the point x₇=7 can be calculated as follow.

$L_{7,7} = {\begin{Bmatrix}{{1l_{1,7}},} \\{{2l_{1,7}},} \\{{3l_{1,7}},} \\{{4l_{1,7}},} \\{{5l_{1,7}},} \\{6l_{1,7}}\end{Bmatrix} = {\begin{Bmatrix}{l_{1,7},} \\{l_{2,7},} \\{l_{3,7},} \\{l_{4,7},} \\{l_{5,7},} \\l_{6,7}\end{Bmatrix} = {\begin{Bmatrix}{{1\lbrack 1234567\rbrack},} \\{{2\lbrack 1234567\rbrack},} \\{{3\lbrack 1234567\rbrack},} \\{{4\lbrack 1234567\rbrack},} \\{{5\lbrack 1234567\rbrack},} \\{6\lbrack 1234567\rbrack}\end{Bmatrix} = \begin{Bmatrix}{1234567,} \\{2461357,} \\{3625147,} \\{4152637,} \\{5316427,} \\6543217\end{Bmatrix}}}}$

By using the (n−1=6)th powder of the cyclic-left-shift operator(t_(l1))⁶ to act on the largest single fixed point subgroup L_(7,7), thefollowing (7,42,6) permutation group code P_(7,7) can be obtained:

$\begin{matrix}\begin{matrix}{P_{7,7} = \left\{ {p_{1},p_{2},\ldots\;,p_{42}} \right\}_{7}} \\{= \left\{ {\left( t_{l\; 1} \right)^{6}L_{7,7}} \right\}} \\{= \left\{ {{\left( t_{l\; 1} \right)^{6}l_{1,7}},{\left( t_{l\; 1} \right)^{6}l_{2,7}},{\left( t_{l\; 1} \right)^{6}l_{3,7}},{\left( t_{l\; 1} \right)^{6}l_{4,7}},{\left( t_{l\; 1} \right)^{6}l_{5,7}},{\left( t_{l\; 1} \right)^{6}l_{6,7}}} \right\}} \\{= \begin{Bmatrix}1234567 & 2461357 & 3625147 & 4152637 & 5316427 & 6543217 \\2345671 & 4613572 & 6251473 & 1526374 & 3164275 & 5432176 \\3456712 & 6135724 & 2514736 & 5263741 & 1642753 & 4321765 \\4567123 & 1357246 & 5147362 & 2637415 & 6427531 & 3217654 \\5671234 & 3572461 & 1473625 & 6374152 & 4275316 & 2176543 \\6712345 & 5724613 & 4736251 & 3741526 & 2753164 & 1765432 \\7123456 & 7246135 & 7362514 & 7415263 & 7531642 & 7654321\end{Bmatrix}}\end{matrix} & \;\end{matrix}$

Example 1 indicates that the code set P_(7,7) is a permutation groupcode with a code length of 7, a minimum distance of 6, a cardinality of42 and an error-correcting capability of 5. In the code set P_(7,7),each column is a coset, which is obtained by storing the firstpermutation of this column in a cyclic shift register and performingcyclic-left-shift operations for n−1=6 times. First permutations in allsix cosets are provided by the largest single fixed point subgroupL_(7,7), and a permutation in L_(7,7) can be calculated by the scalingtransformation f_(a)(l_(1,x) _(i) )=al_(1,x) _(i) .

So far, the codeword enumeration work in the code set of the (n, n(n−1),n−1) permutation group code has been completed by the three methods, andthe coset partition structure characteristics of the code set P_(n,x)_(i) have been described. Generally, a code having the coset partitionstructure characteristics is referred to as a coset code.

Encoding Structure of a General Coset Code

In the traditional coset code, a binary information sequence is actuallymapped to a modulation symbol in the signal set of the constellation,and an encoding method is mainly prescribed. The code set can beregarded as a constellation with a coset partition structure, and eachcodeword can be regarded as a modulation symbol. A binary sequencecarrying information can be mapped to a codeword in the code set byemploying a subset partitioning and prescribing mapping method for allsignal points (i.e., modulation symbols or codewords) in theconstellation. A description of three parts of the encoder for thegeneral coset code is given below by using the lattice and cosetlanguage.

i) An n-dimensional lattice Λ can be seen as an infinite array ofregular points in an n-dimensional space. Signal points may be takenfrom a finite subset of points in a translational coset Λ+a of thelattice Λ, and a set of all possible finite signal points is called asignal constellation.

ii) A finite subset Λ′ of the lattice Λ (i.e., a subset of points of Λ)is itself an n-dimensional sublattice. This sublattice induces apartition expressed as Λ/Λ′, that is, the partition Λ/Λ′ partitions thelattice Λ into |Λ/Λ′| cosets of Λ′, in which |Λ/Λ′| represents an orderof the partition, i.e., a number of the cosets. When Λ and Λ′ are binarylattices, the order of the partition is a power of 2, expressed as 2^(k)¹ , and k=k₁+k₂ represents a length of the information sequence.Accordingly, this partition partitions the signal constellation into2^(k) ¹ subsets, and each subset corresponds to a different coset in Λ′.

iii) A binary encoder C with a code rate of k₁/(k₁+r) inputs k₁ bits pern dimensions, and outputs k₁+r bits. A coset is selected from the |Λ/Λ′|cosets of Λ′ by the k₁+r bits, and a codeword is selected from theselected coset by the remaining uncoded k₂ bits. The redundancy r(C) ofthe encoder C is r bits per n dimensions, and the standard redundancyper two dimensions is ρ(C)=2r(C)/n.

The above three parts constitutes an encoding process of the generalcoset code, that is, mapping a k-bit information digit to a signal point(i.e., a modulation symbol) in the constellation with the cosetstructure characteristics. The coset code can be expressed by a symbol

(Λ/Λ′, C), which represents a set of modulation symbols corresponding toall signal points in the constellation, and is also a set of all binarysequences carrying information corresponding to all signal points in theconstellation. These signal points are located in modulation symbols ofthe cosets of Λ′, and all cosets of Λ′ can be indexed by the encoded bitsequences output by the encoder C. When the encoder C is a linear blockcode,

(Λ/Λ′, C) is called a coset lattice code, and when the encoder C is aconvolutional code,

(Λ/Λ′, C) is called a trellis code.

The redundancy r(C) of the encoder C is r bits per n dimensions, and thestandard redundancy per two dimensions is ρ(C)=2r(C)/n. The basicencoding gain of the coset code is expressed by γ(

), and is defined by two basic geometric parameters: a least squareddistance between two signal points in

(Λ/Λ′, C), and a basic volume V(

) per n dimensions. The volume V(

) is related to the redundancy r(

) of the coset code, and is equal to 2^(r()

⁾. The redundancy r(

) of the coset code is equal to a sum of the redundancy r(C) of theencoder C and the redundancy r(Λ) of the lattice Λ, i.e., r(

)=r(C)+r(Λ). For a regular lattice, r(Λ)=0, and thus, r(

)=r(C)+r(Λ)=r(C). Therefore, the encoding gain of the coset code is:r(

)=d _(min) ²(

)/V(

)=2^(−ρ(C)) d _(min) ²(

)=2^(−2r(C)/n) d _(min) ²(

).

Analysis of the encoding gain r(

) reveals that the redundancy r(C) of the encoder C is always smallerthan the dimension n.

Therefore, the contribution of the redundancy introduced by the encoderC to the encoding gain is to reduce the total gain by 2^(−2r(C)/n)times. The mapping encoding method for the (n, n(n−1), n−1) permutationgroup code based on coset partition proposed in the present disclosuremakes full use of the natural coset partition characteristics of thepermutation group code, eliminating the need for the encoder C to index|Λ/Λ′| cosets in the constellation, so that the complexity of themapping encoding system is reduced (due to the cancellation of theencoder C), and the system gain is improved by 2^(−2r(C)/n) times, orthe reduction of the total gain caused by the encoder C is eliminated.

Technical Solution

The technical solution is partitioned into two parts. The first partcovers an encoding method for a (n, n(n−1), n−1) permutation group codebased on coset partition, and the second part covers a structure designof an encoder for the (n, n(n−1), n−1) permutation group code.

Part 1: An Encoding Method for a (n, n(n−1), n−1) Permutation Group CodeBased on Coset Partition

The present disclosure provides a mapping encoding method for amodulation constellation in a communication system, in which a k-lengthbinary information sequence is mapped to a n-length permutation codewordin a signal constellation Γ_(n) formed by the (n, n(n−1), n−1)permutation group code based on coset partition, that is, a mappingencoding method of mapping a set of 2^(k) k-length binary informationsequences to a set of n(n−1) n-length permutation codewords.

When n is a prime number, a code set P_(n,x) _(i) of a (n, n(n−1), n−1)permutation group code with a code length of n, a cardinality of n(n−1)and a minimum Hamming distance of n−1 is calculated by the followingmethods:

$\begin{matrix}{P_{n,x_{i}} = {{C_{n}L_{n,x_{i}}} = \left\{ {\left. {c_{i} \circ l_{j}} \middle| {c_{i} \in C_{n}} \right.,{l_{j} \in L_{n,x_{i}}},{i \in Z_{n}},{j \in Z_{n - 1}}} \right\}}} & (i) \\{= {\left\{ {\left( t_{rn} \right)^{n - 1}L_{n,x_{i}}} \right\} = \left\{ {\left( t_{l1} \right)^{n - 1}L_{n,x_{i}}} \right\}}} & ({ii})\end{matrix}$where the code set P_(n,x) _(i) has a cardinality of |P_(n,x) _(i)|=n(n−1), and the largest single point subgroup L_(n,x) _(i) iscalculated by the following expression:L _(n,x) _(i) ={a(l _(1,x) _(i) −x _(i))+x _(i) |a∈Z _(n-1) ,x _(i) ∈Z_(n) ,l _(1,x) _(i) =[1 . . . n]}  (iii)where the largest single fixed point subgroup has a cardinality of|L_(n,x) _(i) |=n−1; x_(i)∈Z_(n) indicates that n−1 permutationcodewords in the largest single fixed point subgroup L_(n,x) _(i) allcontain a fixed point x_(i) and the other code elements are non-fixedpoints; and for x_(i)=i and x_(i), i∈Z_(n), there are n fixed points,respectively corresponding to n largest single fixed point subgroupsL_(n,1), L_(n,2), . . . , L_(n,n).

In the expression (i), P_(n,x) _(i) =C_(n)L_(n,x) _(i) indicates thatthe code set P_(n,x) _(i) is composed of n−1 cosets C_(n)l_(1,x) _(i) ,C_(n)l_(2,x) _(i) , . . . , C_(n)l_(n-1,x) _(i) of C_(n). Since in theexpression (i), the composition operations “∘” of the cyclic subgroupC_(n) acting on L_(n,x) _(i) cannot be realized by hardware, C_(n) isreplaced by a cyclic-right-shift composite operator function(t_(rn))^(n-1) or a cyclic-left-shift composite operator function(t_(l1))^(n-1) acting on L_(n,x) _(i) (that is, the cyclic shiftoperation is performed for n−1 times). Thus, n−1 cosets of C_(n) can beequivalently expressed as n−1 orbits {(t_(l1))^(n-1)l_(1,x) _(i) },{(t_(l1))^(n-1)l_(2,x) _(i) }, . . . , {(t_(l1))^(n-1)l_(n-1,x) _(i) }or {(t_(rn))^(n-1)l_(1,x) _(i) }, {(t_(rn))^(n-1)l_(2,x) _(i) }, . . . ,{(t_(rn))^(n-1)l_(n-1,x) _(i) }.

The above equivalent operation of replacing C_(n) with (t_(rn))^(n-1) or(t_(l1))^(n-1) is as follows: firstly, a coset leader or orbit leaderarray composed of n−1 permutation codewords is calculated by theexpression (iii) for calculating L_(n,x) _(i) , and then all n(n−1)codewords of the code set P_(n,x) _(i) are calculated by the expression(ii). For each orbit, a permutation codeword l_(j,x) _(i) of L_(n,x)_(i) is input to a cyclic shift register to perform n−1cyclic-left-shift operations, equivalent to performing{(t_(l1))^(n-1)l_(j,x) _(i) }, or to perform n−1 cyclic-right-shiftoperations, equivalent to performing {(t_(rn))^(n-1)l_(j,x) _(i) },thereby generating n permutation codewords. n(n−1) permutation codewordscan be generated by n−1 orbits.

2^(k) codewords are selected from n(n−1) codewords of the code setP_(n,x) _(i) to form a signal constellation Γ_(n) such that Γ_(n) stillhas the coset characteristics, that is, Γ_(n) contains 2^(k) permutationcodewords, and is partitioned into 2^(k) ¹ ≤n−1 cosets, and each cosetcontains 2^(k) ² ≤n permutation codewords, where k=k₁+k₂, and2^(k)≤n(n−1). Therefore, an exact value of k is k=└log₂ n(n−1)┘.

An encoding method for a one-to-one mapping from the information setH_(k) to the signal constellation Γ_(n) is described as follows. Thereexists a mapping function φ: H_(k)→Γ_(n) defined by a function π=φ(h),by which an information sequence h=[h₁h₂ . . . h_(k)] in a set H_(k) of2^(k) k-length binary information sequences is mapped to a signal pointπ=[a₁a₂ . . . a_(n)] in a signal constellation Γ_(n) composed of 2^(k)n-length permutation codewords, where π∈Γ_(n), h∈H_(k), h₁, h₂, . . . ,h_(k)∈Z₂, a₁, a₂, . . . , a_(n)∈Z_(n).

In summary, the coset code of the (n, n(n−1), n−1) permutation groupcode contains the following three parts.

(A) For any prime number n>1, the code set P_(n,x) _(i) of thepermutation group code is a subgroup of the symmetric group S_(n), andall codewords of P_(n,x) _(i) can be regarded as discrete lattice pointsin an n-dimensional space. All valid signal points are selected from asubgroup or finite lattice P_(n,x) _(i) of the symmetric group S_(n),and a set of the valid signal points is called a signal constellation,which is expressed by Γ_(n)⊂P_(n,x) _(i) and has a cardinality of|Γ_(n)|=|H_(k)|=2^(k). In particular, Γ_(n) and P_(n,x) _(i) have thesame coset structure, and cosets of Γ_(n) have the same size, which issmaller than that of cosets of P_(n,x) _(i) .

(B) For a sub-lattice or sub-group C_(n) of the code set P_(n,x) _(i)(i.e., a subset of n(n−1) codewords of P_(n,x) _(i) ), the sub-latticeC_(n) itself is an n-dimensional lattice or a set of n-dimensionalpermutation vectors each containing n permutation codewords, and thesub-lattice C_(n) induces a partition P_(n,x) _(i) /C_(n) of P_(n,x)_(i) , which partitions P_(n,x) _(i) into |P_(n,x) _(i) /C_(n)| cosetsof C_(n), where |P_(n,x) _(i) /C_(n)|=|L_(n,x) _(i) |=n−1. In the signalconstellation Γ_(n)⊂P_(n,x) _(i) , when a coset of C_(n) and a latticepoint of the coset are respectively indexed by two binary sequences, theorder of this partition can be expressed as a power of 2, the number ofcosets of this division is 2^(k) ¹ , and each coset is indexed by ak₁-length information sequence. In each coset, the number of validlattice points is also a power of 2, then the number of signal pointscontained in each coset is 2^(k) ² , and each signal point in each cosetis indexed by a k₂-length information sequence, where k=k₁+k₂.

The k-length binary information sequence is separated into two binarysequences by a bit splitter D, that is, the bit splitter D inputs ak-length binary information sequence and then outputs two independentbinary sequences: a k₁-length information sequence corresponding tohigh-level k₁-bit of the k-length binary information sequence, and ak₂-length information sequence corresponding to low-level k₂-bit of thek-length binary information sequence. The k₁-length information sequenceis used for indexing a coset in |P_(n,x) _(i) /C_(n)|=|L_(n,x) _(i)|=n−1 cosets, that is, for selecting a coset in the 2^(k) ¹ =n−1 cosetsin Γ_(n). The k₂-length information sequence is used for indexing acodeword in the selected coset, that is, for selecting a codeword in the2^(k) ² <n codewords (signal points) of the selected coset to betransmitted to a channel, thereby obtaining one-to-one mappingnumberings between the 2^(k) k-length binary information sequences andthe |Γ_(n)|=|H_(k)|=2^(k) n-length permutation codewords of the signalconstellation Γ_(n).

In summary, an encoding method of mapping a k-length binary informationsequence to a n-length permutation codeword is obtained by utilizing thenatural coset structure of the (n, n(n−1), n−1) permutation group code,and the coset code is expressed by

(P_(n,x) _(i) /C_(n), D).

Example 1: let n=5, which is a prime number, calculate C₅={c₁, c₂, c₃,c₄, c₅}={12345,23451,34512,45123,51234}, and L_(5,5)={l_(1,5),l_(2,5),l_(3,5), l_(4,5)}={al_(1,5)|a=1,2,3,4;l_(1,5)=[12345]}={12345,24135,31425,43215}. P_(5,5) are obtained byP_(n,x) _(i) ={(t_(l1))^(n-1)L_(n,x) _(i) }.

$P_{5,5} = {\left\{ {{\left( t_{l1} \right)^{4}l_{1,5}},{\left( t_{l1} \right)^{4}l_{2,5}},{\left( t_{l1} \right)^{4}l_{3,5}},{\left( t_{l1} \right)^{4}l_{4,5}}} \right\} = \begin{Bmatrix}12345 & 24135 & 31425 & 43215 \\23451 & 41352 & 14253 & 32154 \\34512 & 13524 & 42531 & 21543 \\45123 & 35241 & 25314 & 15432 \\51234 & 52413 & 53142 & 54321\end{Bmatrix}}$

A binary information sequence with k=4 bits is mapped to a permutationcodeword with a code length of n=5. The coset codes

(P_(5,5)/C₅; D) contains |Γ₅|=|B₄|=2^(k)=2⁴=16 points, which areselected from |P₅|=20 points of P₅. In a feasible numbering method, thebinary information sequence with k=4 bits is separated into a highsignificance digit with k₁=2 bits and a low significance digit with k₂=2bits. k₁=2 bits involve four cases: 00,01,10,11, and thus, 2^(k) ¹ =2²=4cosets C₅l₁, C₅l₂, C₅l₃, C₅l₄ of C₅ in r₅⊂P₅ can be numbered. k₂=2 bitsalso involve four cases: 00,01,10,11, and thus, 2^(k) ² =2²=4permutation codewords in each of the cosets C₅l₁, C₅l₂, C₅l₃, C₅l₄ canbe numbered. Since each coset in C₅ contains 5 permutation codewords,the signal selector in the encoding method is required to discard onepermutation codeword. FIG. 1 is a corresponding relation diagram of abinary information sequence and a permutation codeword in a mappingencoding method according to the present disclosure, in which the lastone permutation codeword in each coset is discarded, and each validconstellation point is matched with a binary information sequence withk=4 bits.

FIG. 2 shows how a codeword 14253 in a numbering table in FIG. 1 isselected by the k=4-bit binary information sequence 1011. The bitsplitter D separates 1011 into two parts: a 2-bit high significant digit10 and a 2-bit low significant digit 11. The third coset C₅l₃ of C₅ inP_(5,5) is selected by the 2-bit high significant digit 10, and thefourth codeword 14253 in the third coset is selected by the 2-bit lowsignificant digit 11. An output of the encoder functions as both aserial port and a parallel port. In this way, the encoder's encodingprocess with an input of 1011 and an output of 14253 is achieved.

Part 2: Structural Design of an Encoder for a (n, n(n−1), n−1)Permutation Group Code Based on Coset Partition

The architecture of the encoder for the (n, n(n−1), n−1) permutationgroup code based on coset partition has two representations, which arerespectively called a basic principle architecture of the encoder and ageneral architecture of an execution circuit of the encoder. Thearchitecture of the execution circuit of the encoder has three circuitexecution schemes: one scheme is that all 2^(k) codewords of the signalconstellation Γ_(n) formed by the code set P_(n,x) _(i) are stored in aread-only memory (ROM), and thus this encoder is called an encoder withall constellation point codewords stored in ROM, which belongs to afirst scheme of the present disclosure (i.e., a U₁-V₁ type encoderdescribed later); and other scheme is that a part of codewords (|P_(n,x)_(i) /C_(n)|=|L_(n,x) _(i) |=n−1 permutation codewords, that is, allcodewords of the largest single fixed point subgroup L_(n,x) _(i) ) ofthe signal constellation Γ_(n) formed by the code set P_(n,x) _(i) arestored in ROM and thus this encoder is called an encoder with a part ofconstellation point codewords stored in ROM. The latter scheme isfurther partitioned into two schemes, i.e., a second scheme of thepresent disclosure and a third scheme of the present disclosure.Specifically, in the second scheme of the present disclosure, n−1permutation codewords of L_(n,x) _(i) are first stored in ROM, thenaccording to the manner of generating respective permutation codewordsby a calculation expression P_(n,x) _(i) ={(t_(rn))^(n-1)L_(n,x) _(i)}={(t_(l1))^(n-1)L_(n,x) _(i) }, a permutation codeword l_(a,x) _(i) isfirst selected from the ROM storing L_(n,x) _(i) with an address mappedby the k₁-length information sequence, and then stored in acyclic-left-shift or cyclic-right-shift register, and the number ofcyclic shift operations performed by the cyclic-left-shift orcyclic-right-shift register is controlled by the k₂-length informationsequence, thereby obtaining an output codeword of the encoder. Thus, inthe second scheme, the encoder can also be called an encoder in which atransmitted codeword is generated by a cyclic shift register (i.e., aU₁-V₂ type encoder described later). In the third scheme, n−1permutation codewords of L_(n,x) _(i) are first generated by the cosetleader array generator by calculating the expression L_(n,x) _(i)={a(l_(1,x) _(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1. . . n]}, then according to the manner of generating respectivepermutation codewords by a calculation expression P_(n,x) _(i)={(t_(rn))^(n-1)L_(n,x) _(i) }={(t_(l1))^(n-1)L_(n,x) _(i) }, apermutation codeword l_(a,x) _(i) is first generated by the coset leaderarray generator (also called an orbit leader array generator) bycalculating l_(a,x) _(i) =a(l_(1,x) _(i) −x_(i))+x_(i), where a isgenerated by the k₁-length information sequence and outputted into acyclic-left-shift or cyclic-right-shift register, and the number ofcyclic shift operations performed by the cyclic-left-shift orcyclic-right-shift register is controlled by the k₂-length informationsequence, thereby obtaining an output codeword of the encoder (i.e., aU₂-V₂ type encoder described later).

A basic principle block diagram of the encoder for the signalconstellation Γ_(n) includes an information sequence splitter D, a cosetselector and an intra-coset permutation codeword selector, as shown FIG.3.

The information sequence splitter D is configured to input a k-lengthbinary information sequence and output two binary information sequences:a k₁-length information sequence corresponding to high-level k₁-bit ofthe input k-length binary information sequence, and a k₂-lengthinformation sequence corresponding to low-level k₂-bit of the inputk-length binary information sequence, where k=k₁+k₂.

The coset selector is configured to select a coset by taking k₁-lengthinformation sequences as indices of n−1 cosets, in which k₁-lengthinformation sequences respectively correspond to 2^(k) ¹ ≤n−1 k₁-lengthbinary index labels, and each k₁-length binary index label is used forselecting a coset from 2^(k) ¹ cosets. When n is a prime number, 2^(k) ¹=n−1, that is, the selection is unique.

The intra-coset permutation codeword selector is configured to select apermutation codeword by taking k₂-length information sequences asindices of n permutation codewords in the selected coset, in which 2^(k)² k₂-length information sequences respectively correspond to 2^(k) ² ≤nk₂-length binary index labels, and each k₂-length binary index label isused for selecting a permutation codeword from selected 2^(k) ²permutation codewords in the selected coset. When n is a prime number,2^(k) ² permutation codewords can be selected from the n permutationcodewords in each coset of C_(n) as needed to form a coset of theconstellation Γ_(n), and any n−2^(k) ² codewords in each coset of C_(n)are required to be discarded, that is, this selection is not unique, buthas multiple solutions.

A general architecture of the mapping encoder for the constellationΓ_(n) includes the following circuit structure, in addition to theinformation sequence splitter D, as shown FIG. 4.

The coset selector has two implementation methods, each of whichconsists of two parts. The first implementation method is called a U₁method, two parts of which include: an address generator of mapping ak₁-length information sequence to a coset leader permutation codeword,in which when k₁ is input, an address in a n-dimensional ROM is output,expressed as k₁→address; and a storage structure of 2^(k) ¹ coset leaderpermutation codewords in the n-dimensional ROM, expressed as 2^(k) ¹codewords→n-dimensional ROM. The second implementation method is calleda U₂ method, two parts of which include: a mapping relationship betweena k₁-length information sequence and a parameter a, expressed ask₁→parameter a; and an orbit leader array generator.

The intra-coset permutation codeword selector has two implementationmethods, each of which consists of two parts. The first implementationmethod is called a V₁ method, two parts of which include: an addressgenerator of mapping a k₂-length information sequence to an intra-cosetpermutation codeword, in which when k₂ is input, an address in an-dimensional ROM is output, expressed as k₂→address; and a storagestructure of 2^(k) ² intra-coset permutation codewords in then-dimensional ROM, expressed as 2^(k) ² codewords→n-dimensional ROM. Thesecond implementation method is called a V₂ method, two parts of whichinclude: a decrement counter for k₂-length information sequence; and acyclic-left-shift or cyclic-right-shift register with two switches.

The two implementation methods U₁ and U₂ of the coset selector and thetwo implementation methods V₁ and V₂ of the intra-coset permutationcodeword selector can be combined to form four different encoders, thatis, U₁-V₁, U₁-V₂, U₂-V₁ and U₂-V₂ type encoders. Among these encoders,the U₂-V₁ type encoder doesn't actually exist due to its contradictorystructure. Specifically, in the V₁ method, 2^(k) ² permutation codewordsin the coset are required to be stored in the n-dimensional ROM, whichrequires the 2^(k) ¹ permutation codewords of the coset leader to befirst stored. However, in the U₂ method, the 2^(k) ¹ permutationcodewords of the coset leader are provided by the orbit leader arraygenerator, but not stored in the n-dimensional ROM. Therefore, threeencoder architectures can be obtained: a U₁-V₁ type encoder with allpermutation codewords of the constellation Γ_(n) stored in an-dimensional ROM; a U₁-V₂ type encoder with a part of permutationcodewords of the constellation Γ_(n) stored in the n-dimensional ROM;and a U₂-V₂ type encoder with the constellation Γ_(n) independent ofn-dimensional ROM.

The mapping encoder architecture with all permutation codewords of theconstellation Γ_(n) stored in the n-dimensional ROM (U₁-V₁ type encoder)includes a k-length information sequence splitter D, an addressgenerator of mapping the k₁-length information sequence to the cosetleader permutation codeword (i.e., k₁→address), an address generator ofmapping the k₂-length information sequence to the intra-cosetpermutation codeword (i.e., k₂→address), and a storage structure of all2^(k) permutation codewords of the constellation Γ_(n) in then-dimensional ROM, as shown in FIG. 5.

For the structure of the address generator of mapping the k₁-lengthinformation sequence to the coset leader permutation codeword, there isa one-to-one correspondence between 2^(k) ¹ k₁-length binary sequencescarrying information and first permutation codewords (called a cosetleader) of the respective cosets, in which the coset leader isdetermined by a largest single fixed point subgroup and thus, n−1 cosetleader (also callled orbit leader) codewords are first calculated byL_(n,x) _(i) ={a(l_(1,x) _(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n),l_(1,x) _(i) =[1 . . . n]}, and then 2^(k) ¹ codewords are selected fromthe n−1 coset leader permutation codewords and stored in then-dimensional ROM to obtain a storage address of each coset leaderpermutation codeword in the n-dimensional ROM. In this way, an addressgenerator that inputs a k₁-length information sequence and outputs anaddress of a selected coset leader permutation codeword is formed. Whenn is a prime number, it can be guaranteed that 2^(k) ¹ =n−1, so that amapping relationship between 2^(k) ¹ k₁-length information sequences andstorage addresses of n−1 coset leader permutation codewords in then-dimensional ROM can be established. In this case, no coset isdiscarded, that is, all n−1 cosets are used in the encoder architecture.

For the structure of the address generator of mapping the k₂-lengthinformation sequence to the intra-coset permutation codeword, aone-to-one correspondence between 2^(k) ² k₂-length information sequenceand permutation codewords in each coset is established. Generally, 2^(k)² <n, that is, each coset of the subgroup C_(n) in the code set P_(n,x)_(i) contains n codewords, each coset of the constellation Γ_(n)contains 2^(k) ² codewords, and thus n−2^(k) ² codewords in each cosetof the code set P_(n,x) _(i) must be discarded to form the constellationΓ_(n) and its coset structure, in which it can be determined accordingto actual needs which n−2^(k) ² codewords in each coset of the subgroupC_(n) are discarded. The obtained 2^(k) ² codewords in each coset of theconstellation Γ_(n) are sequentially stored in a storage unit specifiedby the next storage address of a storage address of the respective cosetleader permutation codeword in the n-dimensional ROM, thereby obtaininga storage address of each permutation codeword in each coset. In thisway, a mapping relationship between 2^(k) ² k₂-length informationsequence and storage addresses of 2^(k) ² codewords in each coset of theconstellation Γ_(n) in the n-dimensional ROM is established. Thus, anaddress generator that inputs a k₂-length information sequence andoutputs an address of a selected intra-coset permutation codeword isformed.

For the storage structure of all 2^(k) permutation codewords of theconstellation Γ_(n) in the n-dimensional ROM, the constellation Γ_(n)contains 2^(k) codewords and is partitioned into 2^(k) ¹ cosets, eachcoset containing 2^(k) ² codeword. Firstly, each codeword of the largestsingle fixed point subgroup is calculated, that is, L_(n,x) _(i)={a(l_(1,x) _(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1. . . n]}, and thus, an address of each coset leader permutationcodeword in the n-dimensional ROM is determined, and these 2^(k) ¹ =n−1addresses are stored in a register of the address generator of mappingthe k₁-length information sequence to the coset leader permutationcodeword to select respective coset leader permutation codewords. Then,2^(k) ² permutation codewords are obtained by a cyclic-left-shift orcyclic-right-shift composition operator acting on each coset leaderpermutation codeword, that is, calculating a set (t_(l1))^(n-1)L_(n,x)_(i) } or {(t_(rn))^(n-1)L_(n,x) _(i) }, and stored in the n-dimensionalROM, with a storage address of the respective coset leader permutationcodeword followed by storage addresses of the respective permutationcodewords. Therefore, the address corresponding to the k₂-lengthinformation sequence is determined by the k₂-length information sequenceitself and the coset leader address.

The mapping encoder architecture with a part of permutation codewords ofthe constellation Γ_(n) stored in the n-dimensional ROM (U₁-V₂ typeencoder) includes a k-length information sequence splitter D, an addressgenerator of mapping the k₁-length information sequence to the cosetleader permutation codeword (i.e., k₁→address), a storage structure of2^(k) ¹ coset leader permutation codewords in the n-dimensional ROM(2^(k) ¹ codewords→n-dimensional ROM), a decrement counter for thek₂-length information sequence, and a cyclic-left-shift orcyclic-right-shift register with two switches, as shown in FIG. 6.

The storage structure of the 2^(k) ¹ coset leader permutation codewordsin the n-dimensional ROM: as shown in FIG. 7, in the storage structureof 2^(k) ¹ permutation codewords of L_(n,x) _(i) in the n-dimensionalROM, each line is a storage word of the n-dimensional ROM, and eachpermutation codeword occupies a storage word. 2^(k) ¹ permutationcodewords are arbitrarily selected from all n−1 coset leader permutationcodewords generated by L_(n,x) _(i) ={a(l_(1,x) _(i)−x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1 . . . n]}, andsequentially stored in the n-dimensional ROM. When n is a prime number,the 2^(k) ¹ =n−1 permutation codewords are stored in the n-dimensionalROM in an order of a=1,2, . . . , n−1. In a case of a read controlsignal Wr=1, upon the arrival of the cp clock pulse, the ROM outputs apermutation codeword in parallel according to a multiple-address codeprovided by an address input signal.

A structure of the decrement counter for the k₂-length informationsequence: a k₂-length information sequence corresponding to low-levelk₂-bit of the k-length information sequence is output to the decrementcounter for the k₂-length information sequence, and the k₂-lengthinformation sequence is assigned to a u register in the decrementcounter to perform a cycle-minus-one operation. When u≠0, the decrementcounter outputs a high-level signal to control the switch 1 to beclosed, and the switch 2 to be opened; and when u=0, the decrementcounter outputs a low-level signal to control the switch 1 to be opened,and the switch 2 to be closed.

Cyclic-left-shift or cyclic-right-shift register: when the switch 1 iscontrolled to be closed, the cyclic shift register performs acyclic-left-shift or cyclic-right-shift operation on the permutationcodeword stored therein to obtain a new permutation codeword, and suchcyclic shift operation is performed for k₂ times, until u is decrementedfrom u≠0 to u=0 through the minus-one operation, thereby forming adecoded codeword in the cyclic shift register. Then, the switch 1 iscontrolled to be opened, and the switch 2 is controlled to be closed, soas to output a serial decoded codeword.

The working process of the encoder with a part of permutation codewordsof the constellation Γ_(n), stored in the ROM is as follows: theinformation sequence splitter D inputs a k-length information sequenceand partitions it into a k₁-length information sequence corresponding tohigh-level k₁-bit and a k₂-length information sequence corresponding tolow-level k₂-bit; the k₁-information sequence is mapped to an address ofa coset leader permutation codeword in the n-dimensional ROM, and theaddress generator outputs the address to select the coset leaderpermutation codeword; the coset leader permutation codeword is input inparallel from the n-dimensional ROM to the n-dimensional cyclic shiftregister through a system bus, and under the control of the decrementcounter for the k₂-length information sequence, a high-level signal isoutput when u≠0, so that the switch 1 is closed and the n-dimensionalcyclic shift register performs a cyclic-left-shift or cyclic-right-shiftoperation; the cyclic-left-shift or cyclic-right-shift operation isperformed once for each cycle-minus-one operation of the decrementcounter, and when u is decremented 0 and a low-level signal is output,the switch 1 is opened and the switch 2 is closed, so that the cyclicshift register stops the cyclic-left-shift or cyclic-right-shiftoperation, but performs a left-shift-output operation to serially outputa decoded codeword.

The mapping encoder architecture with the constellation Γ_(n)independent of n-dimensional ROM (U₂-V₂ type encoder) includes ak-length information sequence splitter D, a mapping of a k₁-lengthinformation sequence to a parameter a, a structure of a coset leaderpermutation codeword generator, a decrement counter for a k₂-lengthinformation sequence, and a cyclic-left-shift or cyclic-right-shiftregister with two switches, as shown FIG. 8.

For a mapping relationship between the k₁-length information sequenceand the parameter a, that is, k₁→parameter a, there is a one-to-onecorrespondence between 2^(k) ¹ k₁-length information sequences andvalues of the parameter a in the calculation expression L_(n,x) _(i)={a(l_(1,x) _(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1. . . n]} of the largest single fixed point subgroup. This calculationexpression is used to generate all n−1 coset leader permutationcodewords, and 2^(k) ¹ permutation codewords are selected from the n−1coset leader permutation codewords to form 2^(k) ¹ coset leaderpermutation codewords of the constellation Γ_(n), that is, k₁→a,a→l_(a,x) _(i) , indicating that a k₁-length information sequence candetermine a coset leader permutation codeword.

The structure of the coset leader permutation codeword generator (i.e.,the orbit leader array generator) has been disclosed in a Chinese Patententitled “CONSTRUCTION METHOD FOR (n, n(n−1), n−1) PERMUTATION GROUPCODE BASED ON COSET PARTITION AND CODEBOOK GENERATOR THEREOF” with theapplication No.: 201610051144.9, or a US Patent entitled “CONSTRUCTIONMETHOD FOR (n, n(n−1), n−1) PERMUTATION GROUP CODE BASED ON COSETPARTITION AND CODEBOOK GENERATOR THEREOF” with the application Ser. No.:15/060,111. A k₁-length information sequence is mapped to a value of a,the value of a and an initial value of (l_(1,x) _(i) −x_(i)) are inputto the orbit leader array generator, and then the orbit leader arraygenerator performs an operation of generating a coset leader permutationcodeword according to the calculation expression L_(n,x) _(i)={a(l_(1,x) _(i) −x_(i))+x_(i)|a∈Z_(n-1), x_(i)∈Z_(n), l_(1,x) _(i) =[1. . . n]}.

The working process of the encoder independent of n-dimensional ROM isas follows: the information sequence splitter D inputs a k-lengthinformation sequence and partitions it into a k₁-length informationsequence corresponding to high-level k₁-bit and a k₂-length informationsequence corresponding to low-level k₂-bit. The k₁-length informationsequence is mapped to the coset leader parameter a∈Z_(n-1), and then thecoset leader permutation codeword generator performs an operation ofgenerating a codeword l_(a,x) _(i) ={a(l_(1,x) _(i) −x_(i))+x_(i)} whenreceiving an initial permutation vector (l_(1,x) _(i) −x_(i)), andoutput a coset leader permutation codeword l_(a,x) _(i) in parallel. Thecodeword l_(a,x) _(i) is output to a cyclic-left-shift orcyclic-right-shift register with two switches. Under the control of thedecrement counter for the k₂-length information sequence, the cyclicshift register performs an operation of generating a decoded codeword,and then serially output a decoded codeword.

It should be readily understood to those skilled in the art that theabove description is only preferred embodiments of the presentdisclosure, and does not limit the scope of the present disclosure. Anychange, equivalent substitution and modification made without departingfrom the spirit and scope of the present disclosure should be includedwithin the scope of the protection of the present disclosure.

What is claimed is:
 1. An encoding method for a (n, n(n−1), n−1)permutation group code in a communication modulation system, wherein theencoding method maps a k-length binary information sequence to an-length permutation codeword in a signal constellation Γ_(n) formed bythe (n, n(n−1), n−1) permutation group code based on coset partition,wherein n is a code length, the encoding method comprising the followingsteps of: constructing the (n, n(n−1), n−1) permutation group code,wherein when n is a prime number, the (n, n(n−1), n−1) permutation groupcode contains n(n−1) permutation codewords, each of the n(n−1)permutation codewords contains n code elements, a minimum Hammingdistance between any two of the n(n−1) permutation codewords is n−1, anda code set P_(n,x) _(i) of the (n, n(n−1), n−1) permutation group codeis obtained by following expressions: $\begin{matrix}{P_{n,x_{i}} = {{C_{n}L_{n,x_{i}}} = \left\{ {{{c_{i} \circ l_{j}}❘{c_{i} \in C_{n}}},{l_{j} \in L_{n,x_{i}}},{i \in Z_{n}},{j \in Z_{n - 1}}} \right\}}} & (1) \\{= {\left\{ {\left( t_{rn} \right)^{n - 1}L_{n,x_{i}}} \right\} = \left\{ {\left( t_{l\; 1} \right)^{n - 1}L_{n,x_{i}}} \right\}}} & (2)\end{matrix}$ wherein the code set P_(n,x) _(i) is obtained by anoperator “∘” composition operation of a special cyclic subgroup C_(n)with a cardinality of |C_(n)|=n and a largest single fixed pointsubgroup L_(n,x) _(i) with a cardinality of |L_(n,x) _(i) |=n−1, c_(i)represents an element in C_(n), l_(j) represents an element in L_(n,x)_(i) , Z_(n) represents a positive integer finite domain expressed byZ_(n)={1,2, . . . , n}, Z_(n-1) represents a positive integer finitedomain expressed by Z_(n-1)={1,2, . . . n−1}, and the code set P_(n,x)_(i) has a cardinality of |P_(n,x) _(i) |=n(n−1); the largest singlefixed point subgroup L_(n,x) _(i) is obtained by an expression (3):L _(n,x) _(i) ={a(l _(1,x) _(i) −x _(i))+x _(i) |a∈Z _(n-1) ,x _(i) ∈Z_(n) ,l _(1,x) ₁ =[1 . . . n]}  (3) wherein when n is a prime number,the largest single fixed point subgroup has a cardinality of |L_(n,x)_(t) |=n−1; x_(i)∈Z_(n) indicates that the i-th code element of each ofall n−1 permutation codewords in the largest single fixed point subgroupL_(n,x) _(i) is a fixed point, and the other code elements are non-fixedpoints; and for x_(i)=i and x_(i), i∈Z_(n), there are n fixed points inall, respectively corresponding to n largest single fixed pointsubgroups L_(n,1), L_(n,2), . . . , L_(n,n)=L_(n); P_(n,x) _(i)=C_(n)L_(n,x) _(i) indicates that the code set P_(n,x) _(i) is composedof n−1 cosets C_(n)l_(1,x) _(i) , C_(n)l_(2,x) _(i) , . . . ,C_(n)l_(n-1,x) _(i) of the special cyclic subgroup C_(n); in theexpression (2), the special cyclic subgroup C_(n) is replaced by acyclic-right-shift operator function (t_(rn))^(n-1) or acyclic-left-shift operator function (t_(l1))^(n-1) to act on the largestsingle fixed point subgroup L_(n,x) _(i) , so that the n−1 cosets of thespecial cyclic subgroup C_(n) are equivalently expressed as n−1cyclic-left-shift orbits {(t_(l1))^(n-1) l_(1,x) _(i) }, {(t_(l1))^(n-1)l_(2,x) _(i) , . . . , {(t_(l1))^(n-1)l_(n-1,x) _(i) } orcyclic-right-shift orbits {(t_(rn))^(n-1) l_(1,x) _(i) }, . . . ,{(t_(rn))^(n-1)l_(2,x) _(i) }, . . . , {(t_(rn))^(n-1)l_(n-1,x) _(i) };for an equivalent operation of {(t_(rn))^(n-1) L_(n,x) _(i) } and{(t_(l1))^(n-1) L_(n,x) _(i) }a coset leader or orbit leader arraycomposed of the n−1 permutation codewords is first calculated by theexpression (3) for calculating the largest single fixed point subgroupL_(n,x) _(i) , and then all n(n−1) permutation codewords of the code setP_(n,x) _(i) are calculated by the expression (2); for each orbit, apermutation codeword l_(j,x) _(i) of the largest single fixed pointsubgroup L_(n,x) _(i) is input to a n-dimension cyclic shift register toperform n−1 cyclic-left-shift operations, equivalent to performing{(t_(l1))^(n-1) l_(j,x) _(i) }, or to perform n−1 cyclic-right-shiftoperations, equivalent to performing {(t_(rn))^(n-1) l_(j,x) _(i) },thereby generating n permutation codewords; the n(n−1) permutationcodewords are generated by the n−1 cyclic-left-shift orbits orcyclic-right-shift orbits; selecting 2^(k) permutation codewords fromthe n(n−1) permutation codewords of the code set P_(n,x) _(i) , to formthe signal constellation Γ_(n) such that the signal constellation Γ_(n)having coset characteristics, that is, the signal constellation Γ_(n)contains the 2^(k) permutation codewords, and is partitioned into 2^(k)¹ cosets, and each coset of the 2^(k) ¹ cosets contains 2^(k) ²permutation codewords, wherein k=k₁+k₂, 2^(k)≤n(n−1), 2^(k) ¹ ≤n−1,2^(k) ² ≤n, and an exact value of k is k=└log₂n(n−1)┘ and satisfies|Γ_(n)|=2^(k); and defining a mapping function φ:H_(k)→Γ_(n) by afunction π=φ(h), and mapping, by the mapping function φ, a k-lengthbinary information sequence h=[h₁h₂ . . . h_(k)] in a set H_(k) of 2^(k)k-length binary information sequences to a signal point π=[a₁a₂ . . .a_(n)] in the signal constellation Γ_(n) composed of the |Γ_(n)|=2^(k)n-length permutation codewords, wherein π∈Γ_(n), h∈H_(k), h₁, h₂, . . ., h_(k)∈Z₂={0,1}, and a₁, a₂, . . . , a_(n)∈Z_(n)={1,2, . . . n}.
 2. Theencoding method according to claim 1, wherein the signal constellationΓ_(n) having the coset characteristics constitutes a coset code, and inthe selecting step: for any prime number n>1, the code set P_(n,x) _(i)is a subgroup of a symmetric group S_(n), and all permutation codewordsof the code set P_(n,x) _(i) are positive integer vectors, which areregarded as discrete lattice points in an n-dimensional real Euclideanspace

^(n); all valid signal points of the signal constellation Γ_(n) areselected from the code set P_(n,x) _(i) of the symmetric group S_(n),wherein the code set P_(n,x) _(i) is the subgroup or finite lattice ofthe symmetric group S_(n); the signal constellation Γ_(n) has acardinality of |Γ_(n)|=|H_(k)|=2^(k), and has the same coset structureas the code set P_(n,x) _(i) , and cosets of the signal constellationΓ_(n) have the same size, which is smaller than the size of the code setP_(n,x) _(i) ; the special cyclic subgroup C_(n) is a sub-lattice orsub-group of the code set P_(n,x) _(i) , and for the special cyclicsubgroup C_(n) of the code set P_(n,x) _(i) , i.e., a subset of then(n−1) permutation codewords of the code set P_(n,x) _(i) , the specialcyclic subgroup C_(n) itself is an n-dimensional lattice or a set ofn-dimensional permutation vectors which are equal to the n permutationcodewords, and induces a partition P_(n,x) _(i) /C_(n) of the code setP_(n,x) _(i) , which partitions the code set P_(n,x) _(i) into |P_(n,x)_(i) /C_(n)| cosets of the special cyclic subgroup C_(n), wherein|P_(n,x) _(i) /C_(n)|=|L_(n,x) _(i) |=n−1; in the signal constellationΓ_(n)⊂P_(n,x) _(i) , when all cosets of the special cyclic subgroupC_(n) and all lattice points of the cosets are respectively indexed bytwo binary information sequences, the order of this partition isexpressed as a power of 2, the number of the cosets of this division is2^(k) ¹ , and each coset is indexed by a k₁-length information sequence;in each coset, the number of valid lattice points is also a power of 2,then the number of signal points contained in each coset is 2^(k) ² ,and each signal point in each coset is indexed by a k₂-lengthinformation sequence, wherein k=k₁+k₂ and 2^(k)=2^(k) ¹ ·2^(k) ² ; andin the defining step: the k-length binary information sequence isseparated into two independent binary information sequences: thek₁-length information sequence corresponding to a high-level k₁-bit ofthe k-length binary information sequence, and the k₂-length informationsequence corresponding to a low-level k₂-bit of the k-length binaryinformation sequence; the k₁-length information sequence is used forindexing a coset in |P_(n,x) _(i) /C_(n)|=|L_(n,x) _(i) |=n−1 cosets,that is, for selecting a coset in the 2^(k) ¹ =n−1 cosets in the signalconstellation Γ_(n); the k₂-length information sequence is used forindexing codewords in the selected coset, that is, for selecting acodeword in the 2^(k) ² <n codewords of the selected coset to betransmitted to a channel, thereby obtaining one-to-one mappingnumberings between the 2^(k) k-length binary information sequences andthe 2^(k) n-length permutation codewords of the signal constellationΓ_(n).
 3. An encoder for a (n, n(n−1), n−1) permutation group code in acommunication modulation system, wherein the encoder maps a k-lengthbinary information sequence to a n-length permutation codeword in asignal constellation Γ_(n) formed by the (n, n(n−1), n−1) permutationgroup code based on coset partition, wherein n is a code length, theencoder comprising: a k-length information sequence splitter Dconfigured to input the k-length binary information sequence and outputtwo information sequences: a k₁-length information sequencecorresponding to a high-level k₁-bit of the input k-length binaryinformation sequence, and a k₂-length information sequence correspondingto a low-level k₂-bit of the input k-length binary information sequence,wherein k=k₁+k₂; a coset selector configured to select a coset by takingthe k₁-length information sequence as indices of n−1 cosets, wherein thek₁-length information sequence respectively corresponds to 2^(k) ¹binary index labels, 2^(k) ¹ ≤n−1, and each binary index label is usedfor selecting the coset from 2^(k) ¹ cosets; when n is a prime number,2^(k) ¹ =n−1; the 2^(k) ¹ cosets are as follows: 2^(k) permutationcodewords are selected from n(n−1) permutation codewords of a code setP_(n,x) _(i) of the (n, n(n−1), n−1) permutation group code to form thesignal constellation Γ_(n) such that the signal constellation Γ_(n)having coset characteristics, that is, the signal constellation Γ_(n)contains the 2^(k) permutation codewords, and is partitioned into the2^(k) ¹ cosets, and each coset contains 2^(k) ² permutation codewords,wherein k=k₁+k₂, 2^(k)≤n(n−1), 2^(k) ¹ ≤n−1, 2^(k) ² ≤n, and an exactvalue of k is k=└log₂n(n−1)┘; and an intra-coset permutation codewordselector configured to select a permutation codeword by taking thek₂-length information sequence as indices of n permutation codewords inthe selected coset, wherein 2^(k) ² k₂-length information sequencesrespectively correspond to 2^(k) ² binary index labels for the npermutation codewords in the selected coset, 2^(k) ² ≤n and each binaryindex label is used for selecting a permutation codeword from the 2^(k)² permutation codewords in the selected coset; when n is a prime number,the 2^(k) ² permutation codewords are selected from the n permutationcodewords in a coset of a special cyclic subgroup C_(n) as needed toform a coset of the signal constellation Γ_(n), that is, any n−2^(k) ²permutation codewords are required to be discarded in each coset of thespecial cyclic subgroup C_(n).
 4. The encoder according to claim 3,wherein the encoder is one of the following three encoders: a U₁-V₁ typeencoder with all permutation codewords of the signal constellation Γ_(n)stored in a first n-dimensional read only memory (ROM), a U₁-V₂ typeencoder with a part of permutation codewords of the signal constellationΓ_(n) stored in a second n-dimensional ROM, and a U₂-V₂ type encoderwith the signal constellation Γ_(n) independent of a third n-dimensionalROM, wherein U₁ and U₂ represent two different types of coset selectors,and V₁ and V₂ represent two different types of intra-coset permutationcodeword selectors; wherein a U₁ type coset selector includes two parts:a first address generator for mapping the k₁-length information sequenceto a coset leader permutation codeword, wherein when k₁ is input, anaddress in a fourth n-dimensional ROM is output; and a storage structureof 2^(k) ¹ coset leader permutation codewords is stored in the fourthn-dimensional ROM; wherein a U₂ type coset selector includes two parts:a mapper for mapping the k₁-length information sequence to a parametera; and a coset leader permutation codeword generator; wherein a V₁ typeintra-coset permutation codeword selector includes two parts: a secondaddress generator for mapping the k₂-length information sequence to anintra-coset permutation codeword, wherein when k₂ is input, an addressin a fifth n-dimensional ROM is output; and a storage structure of 2^(k)² intra-coset permutation codewords is stored in the fifth n-dimensionalROM; and wherein a V₂ type intra-coset permutation codeword selectorincludes two parts: a decrement counter for the k₂-length informationsequence; and a cyclic shift register with two switches configured toperform a cyclic-left-shift or cyclic-right-shift operation.
 5. Theencoder according to claim 4, wherein the U₁-V₁ type encoder includesthe k-length information sequence splitter D, the first addressgenerator for mapping the k₁-length information sequence to the cosetleader permutation codeword, the second address generator for mappingthe k₂-length information sequence to the intra-coset permutationcodeword, and a storage structure of all 2^(k) permutation codewords ofthe signal constellation Γ_(n) is stored in the first n-dimensional ROM;for a structure of the first address generator for mapping the k₁-lengthinformation sequence to the coset leader permutation codeword, there isa one-to-one correspondence between 2^(k) ¹ k₁-length binary informationsequences and coset leaders of respective cosets; each of the cosetleaders is determined by a largest single fixed point subgroup L_(n,x)_(i) , and |L_(n,x) _(i) | coset leader permutation codewords arecalculated by L_(n,x) _(i) ={a(l_(1,x) _(i)−x_(i))+x_(i)|a∈Z_(n-1),x_(i)∈Z_(n),l_(1,x) _(i) =[1 . . . n]}; the2^(k) ¹ coset leader permutation codewords are stored in the fourthn-dimensional ROM, and a storage address of each of the 2^(k) ¹ cosetleader permutation codewords in the fourth n-dimensional ROM isrecorded, thereby forming the first address generator; when n is a primenumber, |L_(n,x) _(i) |=n−1=2^(k) ¹ ; the first address generator formapping the k₁-length information sequence to the coset leaderpermutation codeword inputs the k₁-length information sequence andoutputs an address of a selected coset leader permutation codeword; fora structure of the second address generator for mapping the k₂-lengthinformation sequence to the intra-coset permutation codeword, aone-to-one correspondence between 2^(k) ² k₂-length informationsequences and permutation codewords in each coset is established; then−2^(k) ² permutation codewords are discarded in each coset of the codeset P_(n,x) _(i) , to form the signal constellation Γ_(n); the 2^(k) ²permutation codewords of each coset of the signal constellation Γ_(n)are sequentially stored in the fifth n-dimensional ROM, with a storageaddress of the respective coset leader permutation codeword followed bystorage addresses of the respective permutation codewords, and a mappingfunction between the 2^(k) ² k₂-length information sequences and storageaddresses of the 2^(k) ² permutation codewords of each coset of thesignal constellation Γ_(n) in the fifth n-dimensional ROM isestablished; the second address generator for mapping the k₂-lengthinformation sequence to the intra-coset permutation codeword inputs thek₂-length information sequence and outputs an address of a selectedintra-coset permutation codeword; for a storage structure of all the2^(k) permutation codewords of the signal constellation Γ_(n) in thefirst n-dimensional ROM, the signal constellation Γ_(n) is partitionedinto the 2^(k) ¹ cosets, and each coset contains the 2^(k) ² permutationcodewords; each permutation codeword of the largest single fixed pointsubgroup is calculated, and a storage address of each coset leaderpermutation codeword in the fourth n-dimensional ROM is determined;2^(k) ¹ storage addresses are stored in a register of the first addressgenerator for mapping the k₁-length information sequence to the cosetleader permutation codeword to select respective coset leaderpermutation codewords; when n is a prime number, 2^(k) ¹ =n−1; the 2^(k)² permutation codewords in each coset of the signal constellation Γ_(n)are obtained by a cyclic-left-shift or cyclic-right-shift compositionoperator acting on each coset leader permutation codeword, that is,calculating a set (t_(l1))^(n-1) L_(n,x) _(i) } or {(t_(rn))^(n-1)L_(n,x) _(i) } and stored in the fifth n-dimensional ROM, with thestorage address of the respective coset leader permutation codewordfollowed by the storage addresses of the respective permutationcodewords; an address corresponding to the k₂-length informationsequence is determined by the k₂-length information sequence itself andthe address of the coset leader permutation codeword.
 6. The encoderaccording to claim 4, wherein the U₁-V₂ type encoder includes thek-length information sequence splitter D, the first address generatorfor mapping the k₁-length information sequence to the coset leaderpermutation codeword, the storage structure of the 2^(k) ¹ coset leaderpermutation codewords stored in the fourth n-dimensional ROM, thedecrement counter for the k₂-length information sequence, and the cyclicshift register with the two switches; for the storage structure of the2^(k) ¹ coset leader permutation codewords stored in the fourthn-dimensional ROM, when 2^(k) ¹ permutation codewords of the largestsingle fixed point subgroup L_(n,x) _(i) are stored in the fourthn-dimensional ROM, each line represents a storage word in the fourthn-dimensional ROM, and each permutation codeword occupies a storageword; all of the |L_(n,x) _(i) | coset leader permutation codewords aregenerated by L_(n,x) _(i) ={a(l_(1,x) _(i)−x_(i))+x_(i)|a∈Z_(n-1),x_(i)∈Z_(n),l_(1,x) _(i) =[1 . . . n]}; and the2^(k) ¹ permutation codewords are arbitrarily selected from the |L_(n,x)_(i) | coset leader permutation codewords, and sequentially stored inthe fourth n-dimensional ROM in an order of a=1,2, . . . , |L_(n,x) _(i)|; in a case of a read control signal Rd=1, upon the arrival of a cpclock pulse, the fourth n-dimensional ROM outputs a permutation codeworddesignated by an input address in parallel or serial; for a structure ofthe decrement counter for the k₂-length information sequence, thedecrement counter for the k₂-length information sequence inputs thek₂-length information sequence corresponding to the low-level k₂-bit ofthe k-length binary information sequence, and the k₂-length informationsequence is stored to a u cyclic shift register in the decrement counterto perform a cycle-minus-one operation; when u≠0, the decrement counteroutputs a high-level signal to control a switch 1 to be closed and aswitch 2 to be opened; and when u=0, the decrement counter outputs alow-level signal to control the switch 1 to be opened and the switch 2to be closed; the switch 1 is controlled to perform a cyclic shiftoperation of the cyclic shift register, and the switch 2 is controlledto perform a serial output operation of the cyclic shift register; forthe cyclic shift register, when the switch 1 is controlled to be closed,the cyclic shift register performs a cyclic-left-shift orcyclic-right-shift operation on a permutation codeword stored therein toobtain a new permutation codeword, and such cyclic shift operation isperformed k₂ times, until u is decremented from u≠0 to u=0 through thecycle-minus-one operation, thereby forming a decoded codeword in thecyclic shift register; then the switch 1 is controlled to be opened, andthe switch 2 is controlled to be closed, so as to serially output thedecoded codeword; for a working process of the U₁-V₂ type encoder withthe part of permutation codewords of the signal constellation Γ_(n)stored in the second n-dimensional ROM, the k-length informationsequence splitter D inputs a k-length information sequence andpartitions the k-length information sequence into the k₁-lengthinformation sequence corresponding to the high-level k₁-bit and thek₂-length information sequence corresponding to the low-level k₂-bit;the k₁-length information sequence is mapped to an address of a cosetleader permutation codeword in the fourth n-dimensional ROM, and thefirst address generator outputs the address to select the coset leaderpermutation codeword; the selected coset leader permutation codeword isinput in parallel from the fourth n-dimensional ROM to the cyclic shiftregister through a system bus, and under the control of the decrementcounter for the k₂-length information sequence, the high-level signal isoutput in a case of u≠0, so that the switch 1 is closed and the cyclicshift register performs the cyclic-left-shift or cyclic-right-shiftoperation; the cyclic-left-shift or cyclic-right-shift operation isperformed once for each cycle-minus-one operation of the decrementcounter, until u is decremented 0, and then the low-level signal isoutput, so that the switch 1 is opened, the switch 2 is closed, and thecyclic shift register stops the cyclic-left-shift or cyclic-right-shiftoperation, but performs a left-shift-output operation to serially outputthe decoded codeword.
 7. The encoder according to claim 4, wherein theU₂-V₂ type encoder includes: the k-length information sequence splitterD, the mapper for mapping the k₁-length information sequence to theparameter a, the structure of the coset leader permutation codewordgenerator, the decrement counter for the k₂-length information sequence,and the cyclic shift register with the two switches; for the mapper formapping the k₁-length information sequence to the parameter a, there isa one-to-one correspondence between the 2^(k) ¹ k₁-length informationsequences and values of the parameter a in the calculation expressionL_(n,x) _(i) ={a(l_(1,x) _(i) +x_(i))+x_(i)|a∈Z_(n-1),x_(i)∈Z_(n),l_(1,x) _(i) =[1 . . . n]} of the largest single fixed pointsubgroup; all of the |L_(n,x) _(i) | coset leader permutation codewordsare generated by the above calculation expression, and the 2^(k) ¹permutation codewords are selected from the |L_(n,x) _(i) | coset leaderpermutation codewords to form the 2^(k) ¹ coset leader permutationcodewords of the signal constellation Γ_(n), so that the k₁-lengthinformation sequence determines a coset leader permutation codeword ofthe signal constellation Γ_(n); for the working process of the U₂-V₂type encoder, the k-length information sequence splitter D inputs thek-length information sequence and partitions the k-length informationsequence into the k₁-length information sequence corresponding to thehigh-level k₁-bit and the k₂-length information sequence correspondingto the low-level k₂-bit; the k₁-length information sequence is mapped toa parameter a∈Z_(n-1), and the coset leader permutation codewordgenerator receives an initial permutation vector (l_(1,x) _(i) −x_(i)),performs an operation of generating a coset leader permutation codewordl_(a,x) _(i) =a(l_(1,x) _(i) −x_(i))+x_(i) and then outputs a selectedcoset leader permutation codeword l_(a,x) _(i) in parallel; the cosetleader permutation codeword l_(a,x) _(i) is then output to the cyclicshift register with the two switches, and under the control of thedecrement counter for the k₂-length information sequence, a codewordgeneration process of the encoder, and a decoded codeword is output inparallel or serial.